The safety cap of my above-noted prior patent supports a tubular needle structure for advancement toward, engagement with and movement through the puncturable seal of an associated unit dose medicament vial. The needle construction of the instant invention comprises an improvement over the needle structure disclosed in my prior patent in that conventionally beveled piercing tips of hollow needles constructed of plastic tend to bend upon their engagement with conventional medicament vial puncturable seals and axial thrust upon the needle structure to force the beveled tip through the seal. This bending is a result of the thin plastic wall portion of the sharpened needle tip and is always present. However, certain puncturable seals are tougher than others and an individual seal may have an unusually tough central area thereof to be pierced. Under these circumstances, a seal piercing needle structure constructed of plastic bends sufficiently to impede penetration of the sharpened end of the needle structure through the seal to the extent that the center section of the seal engaged by the needle structure sufficiently resists penetration by the needle structure to be excessively inwardly displaced to the extent that all inward movement of the needle structure to the seal piercing position is insufficient to cause the sharpened tip of the needle structure to complete the seal piercing operation. Of course, if time is a factor in administering the medication and the puncturable seal of the medicament vial from which the medication is to be drawn resists penetration by the associated needle structure serious consequences may result.